The minimal realization from deformation theory
type: Item
instance of: scholarly article
title: The minimal realization from deformation theory
zbMATH Open document ID: 0908.17006
author: Alexander Braverman
author: Anthony Joseph
published in: Journal of Algebra
publication date: 1999-03-14T00:00:00Z
review text: Let \(\mathfrak g\) be a complex simple Lie algebra not of type \(A_n\). Then \(\mathfrak g\) has a unique coadjoint orbit \(\mathcal O\) of minimal nonzero dimension and the enveloping algebra \(U(\mathfrak g)\) of \(\mathfrak g\) has a unique completely prime primitive ideal \(J\) whose associated variety is the Zariski closure \(\overline{\mathcal O}\) of \(\mathcal O\) [Ann. Sci. Éc. Norm. Supér. (4) 9, 1-29 (1976; Zbl 0346.17008)]. The purpose of this paper is to give a realization of \(J\) via Gerstenhaber deformation of the coordinate ring \(A:=S(\mathfrak g)/I\) of the variety \(\overline{\mathcal O}\). This is greatly simplified by the fact that \(A\) is a Koszul algebra, as shown by Bezrukavnikov and Imandar-Mehta. At the same time, the author is able to construct modules for the deformed algebra \(U(\mathfrak g)/J\) and in particular to realize \(J\) in a uniform way as the annihilator of a simple highest weight module. He also recovers an old result of Garfinkle that the graded ideal \(\text{gr }J\) in \(\text{gr }U(\mathfrak g)=S(\mathfrak g)\) coincides with \(I\), using a result of Kostant that \(I\) is generated by quadratic polynomials.
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 1218704
zbMATH Keywords: Gerstenhaber deformation
zbMATH Keywords: minimal nilpotent orbit
zbMATH Keywords: Koszul algebra
zbMATH Keywords: BGG variety
zbMATH Keywords: primitive ideal
zbMATH Keywords: enveloping algebra
zbMATH Keywords: graded ideal
reviewed by: William M. McGovern
MaRDI profile type: Publication
full work available at URL: jabr.1997.7349
OpenAlex ID: W2070636491
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DOI: 10.1006/JABR.1997.7349
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description: scientific article; zbMATH DE number 1218704